Lesson 1: 7.1 Modeling Situations with Differential Equations
Duration of Days: 1
Lesson Objective
Students will be able to translate verbal descriptions of rates of change into first-order differential equations.
Students will understand the relationship between "proportionality" and derivatives, and be able to solve for the constant of proportionality (k) given specific initial conditions.
How do we mathematically represent the phrase "the rate of change of y with respect to t"?
What is the structural difference between a rate being directly proportional vs. inversely proportional to a variable?
How can we use a specific data point (initial condition) to find the exact value of the constant of proportionality?
How does the wording of a word problem dictate whether the derivative is positive or negative (growth vs. decay)?
Differential Equation
Rate of Change
Proportionality
Constant of Proportionality (k)
Directly Proportional
Inversely Proportional
Initial Condition
Independent Variable
Dependent Variable
Exponential Growth/Decay Model
MP1: Make sense of problems and persevere in solving them (translating complex word problems).
MP2: Reason abstractly and quantitatively (moving from physical scenarios like population or cooling to abstract equations).
MP4: Model with mathematics (this is the cornerstone standard for Unit 7).
This lesson focuses on the setup, not the solution. Students are given various real-world scenarios (e.g., a baby's growing shoe size, the decay of a substance, or a cooling cup of coffee). They must identify the variables involved and write an equation involving a derivative that matches the description. A major component involves identifying keywords like "proportional to" and "inversely proportional to" to determine if the variable belongs in the numerator or denominator of the expression.
Purpose: To demonstrate that Calculus is a language used to describe the behavior of the real world. It prepares students for Section 7.6 (Separation of Variables), where they will eventually solve the equations they are building here.
DOK Level 2 (Skill/Concept): Translating simple direct proportionality (e.g., "Rate is proportional to the amount present").
DOK Level 3 (Strategic Thinking): Translating multi-variable scenarios
Support (Scaffolding):Translation Dictionary: Provide a small "Math-to-English" table.
Template Equations: Give students the "skeleton" of a differential equation and have them fill in the blanks.
Extension (Challenge):Newton’s Law of Cooling: Present a problem where the rate is proportional to the difference between two values and ask students to explain why.
Units Analysis: Ask students to determine the units of the constant k based on the units of the other variables in the problem.
AP College Board Assessment